Linear Spherical Sliced Optimal Transport: A Fast Metric for Comparing Spherical Data

TL;DR

This paper introduces LSSOT, a novel efficient metric for comparing spherical probability distributions by embedding them into L^2 spaces through slicing, which preserves geometry and enhances computational speed.

Contribution

The paper proposes the Linear Spherical Sliced Optimal Transport (LSSOT) framework, combining slicing and linear embedding to efficiently compare spherical distributions while maintaining their geometric properties.

Findings

LSSOT is proven to be a valid metric.

LSSOT significantly reduces computation time in applications.

LSSOT achieves high accuracy in practical tasks.

Abstract

Efficient comparison of spherical probability distributions becomes important in fields such as computer vision, geosciences, and medicine. Sliced optimal transport distances, such as spherical and stereographic spherical sliced Wasserstein distances, have recently been developed to address this need. These methods reduce the computational burden of optimal transport by slicing hyperspheres into one-dimensional projections, i.e., lines or circles. Concurrently, linear optimal transport has been proposed to embed distributions into \( L^2 \) spaces, where the \( L^2 \) distance approximates the optimal transport distance, thereby simplifying comparisons across multiple distributions. In this work, we introduce the Linear Spherical Sliced Optimal Transport (LSSOT) framework, which utilizes slicing to embed spherical distributions into \( L^2 \) spaces while preserving their intrinsic geometry, offering a computationally efficient metric for spherical probability measures. We establish the metricity of LSSOT and demonstrate its superior computational efficiency in applications such as cortical surface registration, 3D point cloud interpolation via gradient flow, and shape embedding. Our results demonstrate the significant computational benefits and high accuracy of LSSOT in these applications.